The regulation just says you shall identify the vehicle. A VIN would definitely identify the vehicle. It's just as easy to make a truck number for each vehicle though. Here is a link to the regulation.

(2) Report content. The report shall identify the vehicle and list any defect or deficiency discovered by or reported to the driver which would affect the safety of operation of the vehicle or result in its mechanical breakdown. If no defect or deficiency is discovered by or reported to the driver, the report shall so indicate. In all instances, the driver shall sign the report. On two-driver operations, only one driver needs to sign the driver vehicle inspection report, provided both drivers agree as to the defects or deficiencies identified. If a driver operates more than one vehicle during the day, a report shall be prepared for each vehicle operated.

No, it is not required but I know in ID State they record your DOT number and the truck number at the scale. If they don't see the truck number then they could ask you to come inside and then they will ask you what it is along with 500 other questions. This does not happen every time but it has with me.